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2015

PROPs for Linear Systems

Wadsley, Simon, Woods, Nick

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A PROP is a symmetric monoidal category whose objects are the nonnegative integers and whose tensor product on objects is addition.

A morphism from $m$ to $n$ in a PROP can be visualized as a string diagram with $m$ input wires and $n$ output wires.
For a field $k$, the PROP $\mathrm{FinVect}_k$ where morphisms are $k$-linear maps is used by Baez and Erbele to study signal-flow diagrams.
We aim to generalize their result characterizing this PROP in terms of generators and relations by looking at the PROP $\mathrm{Mat}(R)$ of matrices of values in $R$, where $R$ is a commutative rig (that is, a generalization of a ring where the condition that each element has an additive inverse is relaxed).

PROPs for Linear Systems · Around

Built on

S. Mac Lane, Categorical algebra, Bull. Amer. Math. Soc
1965
Earlier work this paper cites.
M. Sweedler, Hopf Algebras
Original
1969
Earlier work this paper cites.
G. Bergman, Everybody knows what a Hopf algebra is, Amer. Math. Soc. Contemp. Math
1985
Earlier work this paper cites.
T. Pirashvili, On the PROP corresponding to bialgebras, Cahiers Top. Géom. Diff. Cat
2002
Earlier work this paper cites.

Similar

S. Lack, Composing PROPs, Th. Appl. Cat
2004
Cited alongside, same era.
J. Baez, Theorems into Coffee, weblog post at The n-Category Café
2008
Cited alongside, same era.
J. Baez, Theorems into Coffee II, weblog post at The n-Category Café
2008
Cited alongside, same era.
J. Baez and J. Erbele, Categories in control. Available as arXiv:1405.6881
Original
Cited in the paper.
F. Bonchi, P. Sobociński and F. Zanasi, Interacting Hopf algebras. Available as arXiv:1403.7048
Original
Cited in the paper.
M. Markl, Algebras and PROPs. Available as arXiv:0601129
Cited in the paper.

Then

J. Baez, Theorems into Coffee III, weblog post at The n-Category Café
2008
Later among the works it cites.
J. Baez, Theorems into Coffee IV, weblog post at The n-Category Café
2008
Later among the works it cites.
F. Bonchi, P. Sobociński and F. Zanasi, A categorical semantics of signal flow graphs, in CONCUR 2014–Concurrency Theory
2014
Later among the works it cites.

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